Project/Area Number  10640016 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Nagoya University 
Principal Investigator 
NAKANISHI Tomoki Graduate School of Mathematics,Associate Professor, 大学院・多元数理科学研究科, 助教授 (80227842)

CoInvestigator(Kenkyūbuntansha) 
HAYASHI Takahiro Graduate School of Mathematics,Associate Professor, 大学院・多元数理科学研究科, 助教授 (60208618)
OKADA Soichi Graduate School of Mathematics,Associate Professor, 大学院・多元数理科学研究科, 助教授 (20224016)
TSUCHIYA Akihiro Graduate School of Mathematics,Professor, 大学院・多元数理科学研究科, 教授 (90022673)
AOMOTO Kazuhiko Graduate School of Mathematics,Professor, 大学院・多元数理科学研究科, 教授 (00011495)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 2001 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 2000 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  Bethe ansatz / Heisenberg model / quantum group / intergrable models / ベーテ仮説法 / ハイゼンベルグ模型 / 量子群 / 可積分模型 / ベーテ方程式 / ベーテ仮説 / Bethe方程式 / 可積分格子模型 
Research Abstract 
We obtain the following new results on the integrable structure of integrable lattice models. 1. The formal completeness theorem on the Bethe equation for the XXZtype spin models. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.)) By classifying the string solutions of the Bethe equation for the XXZtype spin models in the q=0 limit, we showed that, under the KirillovReshetikhin (KR) conjecture on the KR modules of the quantum affine algebras, the power series formula of the character of the KR module representing the formal completeness of the XXZtype spin models holds for any affine Lie algebra. 2. Reformulation of the KirillovReshetikhin conjecture by the canonical solutions of the Qsystems. (Nakanishi, Kuniba, Tsuboi (Tokyo Univ.)) By observing the principle mechanism for various formulae and theorems obtained in the study of 1, we completely clarified the relation between these power series formulae representing the formal completeness and the underlying functional (algebraic) equations (Qsystem of KRtype). Namely, we introduce a kind of functional equations (finite Qsystem) for a finite number of functions of a finite number of variables. A finite Qsystem has a remarkable property that its unique solution admits two power series formula by the Lagrange inversion formula for several variables. They are exactly a finitevariable analogue of the formal completeness formulae for the XXXtype and XXZtype models. The formal completeness formula can be obtained as the projective limit of this formula.
